Complementary Primes

Expand Messages
• Thought this may be of some interest. First a definition. Let s say we have a prime P. Count the number of digits. Call it N. Multiply the leftmost digit of P
Message 1 of 1 , May 29, 2005
• 0 Attachment
Thought this may be of some interest. First a definition. Let's say we
have a prime P. Count the number of digits. Call it N. Multiply the
leftmost digit of P by 2. Append (N-1) trailing zeros to the result to
make an integer A. If Q=(A-P) is prime then let's call P and Q
"complementary primes". (Best handle I could think of). Here is an
example:

Take the prime P=6121. N=4. 6x2=12. Append (N-1)=3 zeros and get A=12000.
Q=(12000-6121)=5879 which is prime. Then we say P and Q are complementary
primes or CP's. This is the same as writing out 6181 in full positional
notation as (6*10^3)+(1*10^2)+(2*10^1)+(1*10^0) then changing the (+)
operators to (-) operators and calculating the result. Don't know if this
goofy idea is new. Found nothing on the web, but I found that CP's as
defined above do exist in abundance. They suggest some interesting
exercises. Here are three:

(1) Look for longest lists of CONSECUTIVE CP's. The longest I have found
is nine; 34409621, 34409647, 34409651, 34409663, 34409693, 34409719,
34409723, 34409737, 34409759. The associated complementary primes are
25590379, 25590353, 25590349, 25590337, 25590307, 25590281, 25590277,
25590263, 25590241.

(2) Look for Complementary Twin Prime Sets or CTP's. By this I mean you
look for a set of twin primes P and Q which are both CP's and such that
their complementary primes are also twins. The smaller twin of one set is
always complementary to the larger twin of the other set.

The interesting thing about CTP's is that they exist ONLY in well-defined
predictable ranges. They are (3*10^3 to 4*10^3), (6*10^3 to 7*10^3),
(9*10^3 to 10*10^3), (3*10^4 to 4*10^4), (6*10^4 to 7*10^4), (9*10^4 to
10*10^4), (3*10^5 to 4*10^5), (6*10^5 to 7*10^5), (9*10^5 to 10*10^5)
etc.

Here are 2 examples. For the range (3*10^3 to 4*10^3) the two smallest
and the only two CTP sets are (3917, 3919) and (2081, 2083). For the
range (9*10^8 to 10*10^8) the two smallest CPT sets are (900005087,
900005089) and (899994911, 899994913).

(3) Look for sets of TWO CONSECUTIVE CP's with gaps OTHER than 2 and such
that their complementary primes have the SAME gap. In other words
generalize CPT's. This brings out another interesting thing. For gaps
DIVISIBLE by 3, the well-defined ranges mentioned above for CTP's DO NOT
exist. For all gaps NOT DIVISIBLE by 3 the ranges DO exist.

Thanks folks and regards. (Jens this is just your speed! How about a
try?)

Bill Sindelar
Your message has been successfully submitted and would be delivered to recipients shortly.